Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model

Abstract

In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (Ih) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the Ih is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.

Is chaos making a difference? Synchronization transitions in chaotic and nonchaotic neuronal networks

“…The first type of chaotic dynamics in neural systems is typically 14 accompanied by microscopic chaotic dynamics at the level of individual oscillators. The 15 presence of this chaos has been observed in networks of Hindmarsh-Rose neurons [8] and 16 biophysical conductance-based neurons [9][10][11][12]. The second type of chaotic firing pattern 17 is the synchronous chaos.…”

“…http://dx.doi.org/10.1101/224451 doi: bioRxiv preprint first posted online Nov. 24, 2017; a persistent Sodium current (I sd ), a Calcium-activated Potassium current (I sr ) and a 63 hyperpolarization-activated current (I h ). Depending on the parameters, it exhibits a 64 variety of firing patterns including irregular, tonic regular, bursting and chaotic 65 firing [12,32]. Based on the Huber & Braun (HB) thermoreceptor model [11], it will be 66 referred to as the HB+I h model.…”

“…Thus, we developed a parameter sampling procedure that replicated, for both 199 chaotic and non-chaotic populations, a similar firing rate distribution rather than the 200 mean. We also shifted to another parameter sub-space (g sd /g sr ) to take advantage of 201 the complete absence of chaos when g h = 0 (Xu et al [12]). Nevertheless, the .…”

“…41 To cope with this question, in the present paper we studied synchronization 42 transition in heterogeneous networks of interacting neurons. Here we make use of an 43 oscillatory neuron model (HB+I h ) that exhibits either chaotic or non-chaotic behavior 44 depending on biologically plausible parameter regions, as we showed in our previous 45 study [12]. We simulated small-world [31] neural networks consisting on a heterogeneous 46 population of HB+Ih neurons, connected by electrical synapses, and sampling their 47 parameters from either chaotic or non-chaotic regions of the parameter space.…”

“…To do this, we use a model of 154 neural oscillator that can display either chaotic or non-chaotic behavior depending on 155 the parameters (Fig 1, Fig 2A, Xu et al 2017 [12] . CC-BY-NC 4.0 International license peer-reviewed) is the author/funder.…”